Abstract
Abstract stochastic processes have been considered in various contexts by a number of authors. See, for example, Burkholder [2], Da Pratto [3], Kallianpur and Wolpert [14] and Metivier [15]. In this paper we shall examine the structure of H1 and its dual BMO, for martingales taking their values in a Banach space E, and we use this to characterize weakly compact subsets in the former space. These results extend the theory for these Banach spaces developed by Dellacherie, Meyer, Yor and Mokobodzki [5]. The condition imposed on E is that it have the Radon-Nikodym property (RNP), which is not unexpected since this is a necessary and sufficient condition that the martingale convergence theorem holds in E. The connection between RNP and the geometry of E has been under intense study for over fifteen years in functional analysis. However, even without this assumption, by using the theory of lifting [13], a cepresentation theorem for elements in (H 1E )’ is (Theorem 3) (notation is given below). More precisely, every element of (H 1E )’ is of the form \( X \to E\left( {\int {\left\langle {{X_t},d{A_t}} \right\rangle } } \right) \), where the optional process A with integrable variation has E’ as its range.
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© 1986 Birkhäuser Boston, Inc.
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Brooks, J.K., Dinculeanu, N. (1986). H1 and BMO Spaces of Abstract Martingales. In: Çinlar, E., Chung, K.L., Getoor, R.K., Glover, J. (eds) Seminar on Stochastic Processes, 1985. Progress in Probability and Statistics, vol 12. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6748-2_2
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DOI: https://doi.org/10.1007/978-1-4684-6748-2_2
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